/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES After renaming modulo the bijection { a ↦ 0, b ↦ 1, c ↦ 2 }, it remains to prove termination of the 1-rule system { 0 0 1 0 2 0 0 0 ⟶ 0 0 0 0 0 1 0 2 0 0 1 0 2 } Applying sparse tiling TRFC(2) [Geser/Hofbauer/Waldmann, FSCD 2019]. After renaming modulo the bijection { (0,0) ↦ 0, (0,1) ↦ 1, (1,0) ↦ 2, (0,2) ↦ 3, (2,0) ↦ 4, (2,1) ↦ 5, (2,2) ↦ 6, (3,0) ↦ 7 }, it remains to prove termination of the 12-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 0 0 1 2 3 4 0 0 1 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 5 , 0 0 1 2 3 4 0 0 3 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 6 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 1 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 5 , 2 0 1 2 3 4 0 0 3 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 6 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 , 7 0 1 2 3 4 0 0 1 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 5 , 7 0 1 2 3 4 0 0 3 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 6 ↦ 5, 5 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 11-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 0 0 1 2 3 4 0 0 3 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 5 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 1 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 6 , 2 0 1 2 3 4 0 0 3 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 , 7 0 1 2 3 4 0 0 1 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 3 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 6 ↦ 5, 5 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 10-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 1 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 5 , 2 0 1 2 3 4 0 0 3 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 6 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 , 7 0 1 2 3 4 0 0 1 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 5 , 7 0 1 2 3 4 0 0 3 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 6 ↦ 5, 5 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 9-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 3 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 , 7 0 1 2 3 4 0 0 1 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 3 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 6 ↦ 5, 5 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 8-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 , 7 0 1 2 3 4 0 0 1 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 5 , 7 0 1 2 3 4 0 0 3 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 7-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 , 7 0 1 2 3 4 0 0 3 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 6 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7 }, it remains to prove termination of the 6-rule system { 0 0 1 2 3 4 0 0 0 ⟶ 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 2 0 1 2 3 4 0 0 0 ⟶ 2 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 0 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 3 ⟶ 4 0 0 0 0 1 2 3 4 0 1 2 3 6 , 7 0 1 2 3 4 0 0 0 ⟶ 7 0 0 0 0 1 2 3 4 0 1 2 3 4 } Applying the dependency pairs transformation. Here, ↑ marks so-called defined symbols. After renaming modulo the bijection { (0,↑) ↦ 0, (0,↓) ↦ 1, (1,↓) ↦ 2, (2,↓) ↦ 3, (3,↓) ↦ 4, (4,↓) ↦ 5, (2,↑) ↦ 6, (4,↑) ↦ 7, (5,↓) ↦ 8, (6,↓) ↦ 9, (7,↑) ↦ 10, (7,↓) ↦ 11 }, it remains to prove termination of the 64-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 8 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 9 , 10 1 2 3 4 5 1 1 1 ⟶ 10 1 1 1 1 2 3 4 5 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 10 1 2 3 4 5 1 1 1 ⟶ 7 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 3 1 2 3 4 5 1 1 1 →= 3 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 , 11 1 2 3 4 5 1 1 1 →= 11 1 1 1 1 2 3 4 5 1 2 3 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 55-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 8 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 9 , 10 1 2 3 4 5 1 1 1 ⟶ 10 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 3 1 2 3 4 5 1 1 1 →= 3 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 , 11 1 2 3 4 5 1 1 1 →= 11 1 1 1 1 2 3 4 5 1 2 3 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 54-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 8 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 9 , 10 1 2 3 4 5 1 1 1 ⟶ 10 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 , 11 1 2 3 4 5 1 1 1 →= 11 1 1 1 1 2 3 4 5 1 2 3 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9, 10 ↦ 10, 11 ↦ 11 }, it remains to prove termination of the 44-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 6 4 5 , 7 1 2 3 4 5 1 1 1 ⟶ 7 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 1 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 5 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 7 1 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 8 , 7 1 2 3 4 5 1 1 2 ⟶ 6 4 8 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 5 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 7 1 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 9 , 7 1 2 3 4 5 1 1 4 ⟶ 6 4 9 , 10 1 2 3 4 5 1 1 1 ⟶ 10 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 , 11 1 2 3 4 5 1 1 1 →= 11 1 1 1 1 2 3 4 5 1 2 3 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 2: 0 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 1 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ 11 ↦ ⎛ ⎞ ⎜ 1 0 ⎟ ⎜ 0 1 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 7 ↦ 6, 8 ↦ 7, 9 ↦ 8, 10 ↦ 9, 11 ↦ 10 }, it remains to prove termination of the 36-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 2 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 6 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 6 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 8 , 9 1 2 3 4 5 1 1 1 ⟶ 9 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 7 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 10 1 2 3 4 5 1 1 1 →= 10 1 1 1 1 2 3 4 5 1 2 3 4 5 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 10 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 35-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 2 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 6 1 2 3 4 7 , 6 1 2 3 4 5 1 1 2 ⟶ 0 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 6 1 2 3 4 8 , 6 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 8 , 9 1 2 3 4 5 1 1 1 ⟶ 9 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 7 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8, 7 ↦ 9 }, it remains to prove termination of the 28-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 4 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 0 1 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 5 1 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 6 1 2 3 4 7 , 6 1 2 3 4 5 1 1 4 ⟶ 0 2 3 4 7 , 8 1 2 3 4 5 1 1 1 ⟶ 8 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8, 7 ↦ 9 }, it remains to prove termination of the 21-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 7 ↦ 7, 8 ↦ 8, 9 ↦ 9 }, it remains to prove termination of the 12-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 , 7 1 2 3 4 5 1 1 1 ⟶ 7 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 9 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 9 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 0 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 6 ↦ 6, 8 ↦ 7, 9 ↦ 8 }, it remains to prove termination of the 11-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 0 1 2 3 4 5 1 1 1 ⟶ 6 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 1 1 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 0 1 2 3 4 5 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 1 2 3 4 5 , 6 1 2 3 4 5 1 1 1 ⟶ 6 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 7 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 8 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 8 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 6 ↦ 0, 1 ↦ 1, 2 ↦ 2, 3 ↦ 3, 4 ↦ 4, 5 ↦ 5, 7 ↦ 6, 8 ↦ 7 }, it remains to prove termination of the 5-rule system { 0 1 2 3 4 5 1 1 1 ⟶ 0 1 1 1 1 2 3 4 5 1 2 3 4 5 , 1 1 2 3 4 5 1 1 1 →= 1 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 1 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 5 , 5 1 2 3 4 5 1 1 2 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 6 , 5 1 2 3 4 5 1 1 4 →= 5 1 1 1 1 2 3 4 5 1 2 3 4 7 } The system was filtered by the following matrix interpretation of type E_J with J = {1,...,2} and dimension 10: 0 ↦ ⎛ ⎞ ⎜ 1 0 1 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 1 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 1 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 1 0 0 0 0 0 1 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 1 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 2 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 1 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 3 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 1 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 4 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 1 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 5 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 1 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 6 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ 7 ↦ ⎛ ⎞ ⎜ 1 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 1 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 ⎟ ⎝ ⎠ After renaming modulo the bijection { 1 ↦ 0, 2 ↦ 1, 3 ↦ 2, 4 ↦ 3, 5 ↦ 4, 6 ↦ 5, 7 ↦ 6 }, it remains to prove termination of the 4-rule system { 0 0 1 2 3 4 0 0 0 →= 0 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 0 →= 4 0 0 0 0 1 2 3 4 0 1 2 3 4 , 4 0 1 2 3 4 0 0 1 →= 4 0 0 0 0 1 2 3 4 0 1 2 3 5 , 4 0 1 2 3 4 0 0 3 →= 4 0 0 0 0 1 2 3 4 0 1 2 3 6 } The system is trivially terminating.